Random walks on random walks: non-perturbative results in high dimensions

Abstract: Consider the dynamic environment governed by a Poissonian field of independent particles evolving as simple random walks on $\mathbb{Z}^d$. The random walk on random walks model refers to a particular stochastic process on $\mathbb{Z}^d$ whose evolution at time $t$ depends on the number of such particles at its location. We derive classical limit theorems for this instrumental model of a random walk in a dynamic random environment, applicable in sufficiently high dimensions. More precisely, for $d \geq 5$, we prove a strong law of large numbers and large deviation estimates. Further, for $d\geq 9$, we obtain a functional central limit theorem under the annealed law. These results are non-perturbative in the sense that they hold for any positive density of the Poissonian field. Under the aforementioned assumptions on the dimension they therefore improve on previous work on the model. Moreover, they stand in contrast to the anomalous behaviour predicted in low dimensions.